Question 119087
Remember: 
to be {{{congruent}}} is not same as to be {{{ similar}}} 

The {{{median}}} of a triangle is a {{{line}}} from a {{{vertex}}} to the {{{midpoint}}} of the opposite side.

 
Given:

Triangles {{{ABC }}}and {{{PQR}}}

{{{AB}}} proportional to {{{PQ}}}

{{{BC}}} proportional to {{{QR}}}
 
and median {{{AD}}} proportional to {{{PM}}}


to prove:

triangles{{{ABC }}}and {{{PQR}}} are similar

{{{ABC}}}~{{{PQR}}}

proof:

prove that three sides of triangles are proportional

since {{{M}}} is midpoint of {{{AC}}}, then	

{{{AD}}} is proportional to {{{PM}}} and {{{DC}}} proportional to {{{MR}}}

since {{{AD + DC = AC}}} 

and {{{PM + MR = PR}}}	

we can conclude that {{{AC}}} is proportional to {{{PR}}}

two triangles {{{ABC}}} and {{{PQR}}} have three sides that are proportional

{{{AB}}} proportional to {{{PQ}}}

{{{AC}}} proportional to {{{PR}}}

and {{{BC}}} proportional to {{{QR}}}

therefore triangles{{{ABC}}} and {{{PQR}}}  are {{{similar }}}, or 

{{{ABC}}}~{{{PQR}}}